f function 56. If f is homogeneous of degree n, show that af 2012 z +...
Suppose that k e N and that f R"-R is homogeneous of order k: that is, that 'f(px)- kf(x) for all x є Rn and all є R. If f is differentiable on R", prove that af Экп af axi (Xi , . . . , xn) ER". for all x
Suppose that k e N and that f R"-R is homogeneous of order k: that is, that 'f(px)- kf(x) for all x є Rn and all є R. If...
Show that if the production function is homogeneous of degree 2, then the cost minimizing demand for input i is given by: xi = h(w) √y, where y and w are output and input prices and h(w) is some function of input prices.
a = 5 , b = 8
Let a and b the last two digits of your QUID. Let f a differentiable function that satisfies the equation tºf(tx,ty)= tf(x,y), for t > 0 Show that f satisfies the following partial differential equation and find the value of k af =kf ax X +y af - ду
Question 4 (20 points) Let F: R R be any homogeneous polynomial function (with degree no less than one) with at least one positive value. Prove that the function f:Rn R, f(x) F(x) 1, defines on f-1(0) a structure of smooth manifold.
Find the constrained extrema of the function f (x, y, z) = x + y + z on the plane given by the equation x^2 + 2xy + 2y^2 + 3z^2 = 1.
(5) Show that if t,... ., are positive numbers such that ty+..+tn-1 and .. n E (0,1/2], then n. First check that the function f(x) In (1-)-n(x) is convex on the interval (0, 1/2). The inequality is due to Ky Fan.]
(5) Show that if t,... ., are positive numbers such that ty+..+tn-1 and .. n E (0,1/2], then n. First check that the function f(x) In (1-)-n(x) is convex on the interval (0, 1/2). The inequality is due to...
(1 point In general for a non-homogeneous problem y' + p(x) +(z) = f() assume that y. is a fundamental set of solutions for the homogeneous problemy" p(x) + (2) 0. Then the formula for the particular solution using the method of variation of parameters is where (z)/ and ()/() where W() is the Wronskian given by the determinant W (2) (2) W2) 31(2)/(2) dr. NOTE When evaluating these indefinite integrals we take the W(2) So we have the de...
need help with all a, b, c
2. 15 Marks (a) Suppose that f : R" R is convex but not necessarily smooth. Prove that h-af is a (b) Suppose that f : R -R is convex and smooth. Also assume that f(x) > 0 for all z (c) Show that the set S = {(x,y) : y > 0} is convex and that the function f(x,y)-x2/v is convex function if a-0. Show with a simple example that this is...
For the following functions, answer if the function is
homogeneous in (x , y), and if yes, what degree it is.
a.
b.
c.
Does any of these three production functions display constant
returns to scale?
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2 Homogeneous coordinates Recall that an affine function is of the form f^x) Mx + t for a matrix M and vector t. Homogeneous coordinates are frequently used to represent affine functions in robotics and 3D graphics. We define the function H by and if f-x) Mxtt where then C0 a. Some vectors are valid homogeneous representations of vectors, and some are not. Explain how to tell if some vector y-0 is the homogeneous representation of some other vector -y...